Example 1

Why can an obtuse triangle have only one obtuse angle?

Hint

Use the triangle Angle Sum Theorem. Or try fitting two elephants into a one-bedroom flat.

Answer

Let's see what happens if we say that ∠1, ∠2, and ∠3 make up the interior angles of a triangle and both ∠1 and ∠2 are obtuse. Obtuse angles are greater than 90°, so just adding the measures of ∠1 and ∠2 would be greater than 180. That means ∠3 would have to be negative and we can't have negative angles.

Example 2

Why can a right triangle have only one right angle?

Hint

Two right angles equal how many degrees? A triangle equals how many degrees?

Answer

Suppose we have 2 right angles in a triangle. The sum of two right angles is already 180°. If we use the Angle Sum Theorem, our remaining angle is 0°, which is impossible.

Example 3

Are all equilateral triangles isosceles?

Hint

What are the definitions of equilateral triangles and isosceles triangles?

Answer

Yes, equilateral triangles are always isosceles.

Example 4

Are all isosceles triangles equilateral?

Hint

In this case, "all isosceles triangles" means, "any and every variation of isosceles triangles imaginable." But you already knew that.

Answer

No, isosceles triangles are not always equilateral. Only if all three sides of the isosceles triangle have the same length would it be equilateral.

Example 5

A triangle with differing side lengths and angles of 82°, 92° and 6° is what kind of triangle?

Hint

We can classify angles based on side lengths and angles.

Answer

The triangle is obtuse and scalene.

Example 6

A triangle has at least two side lengths that are equal to each other and all three angles equaling 60°. What kind of triangle is it?

Hint

What kind of a triangle has angles that are all equal to each other?

Answer

The triangle is equilateral. It could also be described as an acute isosceles triangle, but equilateral is the most accurate description.